My textbook does the proof in a different way. Here's my attempt:
Since $f,g$ is integrable on $[a,b]$ then there is a family of partitions $\{P_n\}$ and $\{Q_n\}$ such that
$$\lim_{n\to\infty} L(P_{n},f) = \lim_{n\to\infty} U(P_{n}, f)$$ $$\lim_{n\to\infty} L(Q_{n},g) = \lim_{n\to\infty} U(Q_{n}, g)$$
Let $R_{n} = P_{n} \cup Q_{n}$. Since $R_{n}$ is a refinement of $P_{n}$ the following holds :
$$L(P_{n},f) \le L(R_{n},f) \le U(R_{n}, f) \le U(P_{n}, f)$$. By the squeeze theorem, we have that $$\lim_{n\to\infty} L(R_{n},f) = \lim_{n\to\infty} U(R_{n}, f)$$
And similarly
$$\lim_{n\to\infty} L(R_{n},g) = \lim_{n\to\infty} U(R_{n}, g)$$
We note that
$$L(R_n , f) + L(R_n , g) \le L(R_n , f+g) \le U(R_n , f+g) \le U(R_n , f) + U(R_n , g)$$
By the squeeze theorem, we have that $\lim_{n\to \infty} L(R_n , f+g) = \lim_{n\to \infty} U(R_n , f+g) = \lim_{n\to \infty} L(R_n , f) +\lim_{n\to \infty} L(R_n , g)$. Hence it follows that $\int_{a}^{b}( f+g) = \int_{a}^{b} f + \int_{a}^{b} g$
Is this proof correct? Here's the textbook proof:
We do not, in general, have that $L(f,P)+L(g,P)=L(f+g,P)$ or that $U(f,P)+U(g,P)=U(f+g,P)$ - the infs and sups can happen in different places between the two functions. When you combine the functions, there needs to be an inequality there. After all, it's possible for the sum of functions to be integrable even if neither of the original functions were. By not addressing this, your argument fails.
Now that the argument has been edited, it works.